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Last edited 2/17/2016 4:32:00 AM
THE QUOTIENT OF A FUNCTION
Any function has an equivalence kernel, which is an equivalence relation on
the domain, and a quotient set, a partition on the domain. The equivalence kernel and the quotient set correspond to each other according to the Fundamental
Theorem on equivalence relations. An important example of this pair of concepts
is the kernel and its set of cosets of a group homomorphism.
Definition
Let F : A ® B be a function. The equivalence kernel of F is the equivalence relation
EF on A determined by requiring that aEF a¢if and only if F(a) = F(a ') . The quotient
set of F is the partition P of A determined by the Fundamental Theorem on equivalence
relations, which means that the block [a ]P is the set of all a ' Î A for which F(a ') = F(a) .
Terminology
The quotient set of a function F : A ® B may be denoted S/F and may be called the
quotient of F. This terminology is not common, and neither is the notation EF for the
equivalence kernel.
Example
Let S be the set {1, 2, 3, 4, 5, 6, 7} . Let F : S ® Z be the function shown in the picture. Its quotient set is obtained by grouping the elements of S into sets according to
their value under F. In this example, S / F = {{1}, {2,7},{3,4,5},{6}} . It is a partition, with
blocks {1}, {2, 7}, {3, 4, 5} and {6}.
1
1
2
7
3
2
4
4
5
3
6
5
6
7
8
9
..
.
2
Other examples
 The quotient set of the squaring function x a x 2 : R ® R is the set
{{x,- x}: x Î R }. Note that every block except one has two elements. For example,
{2, 2} and {p , p }are blocks. The only exception is the block {0}, which has only one element since 0 = –0.
 The quotient set of an injective function consists of singletons. It is worth doing
to play with examples until this becomes obvious!
 Let h : G ® H be a homomorphism of groups with kernel K. Then the quotient
set of h is the set of cosets of K. In this case the quotient set is denoted by G/K, which
is not ambiguous because once you know K (which must be a normal subgroup) you
know all the cosets of K. The key fact here is that G/K has a natural group structure
since the equivalence kernel determined by h is a congruence. This idea occurs in
many other categories of mathematical structures.
References
Algebra, by Saunders MacLane and Garrett Birkhoff, American Mathematical Society,
1999. ISBN 0821816462, 9780821816462
Charles Wells, Class Notes for Discrete Mathematics, 1999.